The real images really exist. The virtual image exist in the brain of the observer.
More seriously: In case of real image, the rays emerging from a point of an object become convergent after passing the imaging system (mirror or lens) and intersect in the same point: at the image of the object.
In case of virtual image the rays emerging from a point of the object are divergent after leaving the lens or mirror. The divergent rays arrive to the eye and the brain interprets them as coming from a common point - from the image. But this common point does not exist in reality.